Dan's answer about which frequencies tend to be used for emergency communication is good, but I'd like to add another aspect you seem to be overlooking.

Don't dismiss the possibility of such communications being done outside of the amateur bands.

In the US (which appears to be the focus of this question because of its reference to Technician and General class licenses), there is MARS (the Military Auxiliary Radio System)), licensing for which is legally completely separate from amateur radio and hence for which there exists a different set of requirements. Operating on HF MARS frequencies would not be influenced by whether or not the operator holds a HF-allowed amateur radio license, or indeed any amateur radio license at all.

Other countries have similar systems in place.

For example, in Sweden, to the extent that volunteers participate in such communications most communication is done on frequencies allocated for civilian defense or military defense purposes generally through the Voluntary Radio Organization (FRO) (not to be confused with the Swedish National Defence Radio Establishment, FRA). I think it's safe to say that many countries that do it that way lean toward allocating such frequencies near the amateur radio bands to (at least as an unspoken goal) ease interoperability with off-the-shelf amateur radio equipment, but there is certainly no requirement that it is done that way.

That said, one should never dismiss the value of the practice one can gain on the amateur radio bands. Particularly on HF, amateur radio communications is often fraught with issues: you're working with relatively low power, possibly less-than-ideal antennas, sometimes lots of stations transmitting within your receiver standard passband, potentially high noise levels, fading, perhaps running off battery power, and so on and so forth. Learning to consistently pick a single station's signal out of all that when times are easy is invaluable practice for being able to assist when the conditions are worse (loss of normal communication links and mains power, for example, forcing reliance on a noisy HF link for regional communications and battery power which limits the available output power).

That signal processor between your ears needs a long training period and constant tuning to be able to pick out signals most effectively.

Dan's answer about which frequencies tend to be used for emergency communication is good, but I'd like to add another aspect you seem to be overlooking.

Don't dismiss the possibility of such communications being done outside of the amateur bands.

In the US (which appears to be the focus of this question because of its reference to Technician and General class licenses), there is MARS (the Military Auxiliary Radio System), licensing for which is legally completely separate from amateur radio and hence for which there exists a different set of requirements. Operating on HF MARS frequencies would not be influenced by whether or not the operator holds a HF-allowed amateur radio license, or indeed any amateur radio license at all.

Other countries have similar systems in place.

For example, in Sweden, to the extent that volunteers participate in such communications most communication is done on frequencies allocated for civilian defense or military defense purposes generally through the Voluntary Radio Organization (FRO) (not to be confused with the Swedish National Defence Radio Establishment, FRA). I think it's safe to say that many countries that do it that way lean toward allocating such frequencies near the amateur radio bands to (at least as an unspoken goal) ease interoperability with off-the-shelf amateur radio equipment, but there is certainly no requirement that it is done that way.

That said, one should never dismiss the value of the practice one can gain on the amateur radio bands. Particularly on HF, amateur radio communications is often fraught with issues: you're working with relatively low power, possibly less-than-ideal antennas, sometimes lots of stations transmitting within your receiver standard passband, potentially high noise levels, fading, perhaps running off battery power, and so on and so forth. Learning to consistently pick a single station's signal out of all that when times are easy is invaluable practice for being able to assist when the conditions are worse such as during a loss of normal communication links and mains power, forcing reliance on a noisy HF link for regional communications and quite possibly battery power which limits the available output power.

That signal processor between your ears needs a long training period and constant tuning to be able to pick out signals most effectively.